At each flow time \(t\) , we obtain a new integration contour. The holomorphic gradient flow is often applied starting from the real axis. In this case, at early flow time, the holomorphic gradient flow is guaranteed to improve the sign problem, as is discussed in [1]}. This can be seen by looking at how the action changes with flow time:
\(\frac{\mathrm {d} S}{\mathrm {d} t} = \left|\frac{\partial S}{\partial \phi }\right|^2\text{.}\) 
